The temperature dependence of the R magnetization and that of the R anisotropy constants were calculated using the molecular field approach, with values of anisotropy constants derived fr[r]
(1)Original article
Calculation of the magnetic properties of pseudo-ternary R2M14B
intermetallic compounds (R ¼ rare earth, M ¼ Fe, Co)
Gabriel Gomez Eslavaa,b,*, Masaaki Itoc, Masao Yanoc, Nora M Dempseya,b, Dominique Givorda,b,d
aUniv Grenoble Alpes, Inst NEEL, F-38000 Grenoble, France bCNRS, Inst NEEL, F-38000 Grenoble, France
cAdvanced Material Engineering Div., Toyota Motor Corporation, Susono 410-1193, Japan dInstituto de Fisica, Universidade Federal Rio de Janeiro, Rio de Janeiro, Brazil
a r t i c l e i n f o
Article history: Received 14 June 2016 Accepted 14 June 2016 Available online 18 June 2016 Keywords:
Molecularfield calculations Crystalline-electricfield interactions R2Fe14B intermetallic compounds
NdFeB magnets
a b s t r a c t
The extrinsic properties of NdFeB-based magnets can be tuned through partial substitution of Nd by another rare-earth element and Fe by Co, as such substitution leads to a modification in the intrinsic properties of the main phase Optimisation of a magnet's composition through trial and error is time consuming and not straightforward, since the interplay existing between magnetocrystalline anisotropy and coercivity is not completely understood In this paper we present a model to calculate the intrinsic magnetic properties of pseudo-ternary Nd2Fe14B-based compounds As concrete examples, which are
relevant for the optimisation of NdFeB-based high-performance magnets used in (hybrid) electric ve-hicles and wind turbines, we consider partial substitution of Nd by Dy or Tb, and Fe by Co
© 2016 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)
1 Introduction
Today's high performance magnets are based on the Nd2Fe14B phase [1,2] Partial substitution of Nd by another rare-earth (R) element, and/or Fe by Co, leads to a change in the intrinsic magnetic properties of the main phase This in turn leads to a change in the extrinsic properties of the magnet Such partial substitution may be motivated by the desire to improve a given intrinsic property (e.g addition of Dy to increase the anisotropyfield and thus the coer-civity, addition of Co to increase the Curie temperature), or to reduce the use of a given element (e.g addition of Ce, which is more abundant and thus cheaper than Nd), for economic and strategic reasons The intrinsic properties of R2M14B (M¼ Fe or Co) have been modelled using a molecularfield approach for the exchange interactions and a single-ion model for the crystalline-electricfield (CEF) interactions[3,4] We recently presented a classical mean-field approach to calculate the temperature dependence of the magnetization and anisotropy of a series of R2M14B compounds[5] Relatively good agreement was found with experimental values from literature achieved with single crystals Here we have
extended this approach to calculate the properties of
ðR1xR0xÞ2ðFe1yCoyÞ14B compounds Such calculations may be used in the analysis of experimentally determined magnetic properties of such compounds and to guide the optimisation of magnet development
2 Molecularfield and CEF coefficients in R2M14B compounds The magnetic properties of R2M14B compounds were exten-sively studied at the end of the 1980's [1,2] To a good first
approximation, they can be described within a mean-field
approach, in which the magnetic properties of the Fe sublattice are essentially taken as identical to those of the R2M14B compounds with non-magnetic R elements The magnetic behaviour of the R elements depends on R-M exchange interactions and on CEF in-teractions with the surrounding electrical charges[3,4] The ReR interactions are very weak and can be neglected[6] The R-M ex-change interactions, described in the meanfield approach, depend on one molecularfield coefficient nRM, which can be written as nRMẳ n0RMẵ2gJ 1ị=gJ, where gJis the Lande factor, the value of which depends on the R element The term between brackets ex-presses the fact that the interactions are between spin moments Exchange interactions between two 4f rare-earth moments are indirect, mediated by 5d electrons The on-site 5de4f interactions decrease from the beginning of the lanthanide series to the end, essentially because the distance between the 5d and the 4f shell
* Corresponding author CNRS, Inst NEEL, F-38000 Grenoble, France E-mail address:grgomeze@gmail.com(G Gomez Eslava)
Peer review under responsibility of Vietnam National University, Hanoi
Contents lists available atScienceDirect
Journal of Science: Advanced Materials and Devices j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j s a m d
http://dx.doi.org/10.1016/j.jsamd.2016.06.014
(2)increases due to the“lanthanide contraction” effect[7] As a result, the coefficient n0
RM(and consequently nRM) is not a constant across the series but varies from one R element to the next The value of the coefficient nRMin each R2M14B compound has been derived from that of the Curie temperature, TCin Ref.[7]for M¼ Fe and in Ref.[6]for M¼ Co
The CEF interactions depend on a limited number of parame-ters, determined by the symmetry of the crystal structure In the present case, the CEF Hamiltonian takes the form:
HCEFẳ B02O02ỵ B2;s2 O2;s2 ỵ B04O40ỵ B4;c4 O4;c4 ỵ B40O04ỵ B4;c6 O4;c6 (1)
where the Om
n are the Stevens coefficients and the Bmn the associated CEF parameters (here, the index n represents the order of the co-efficient and the index m obey the rules m < n and m < 4) The Bm n may be re-expressed asqnAmn< rn> whereqnis a coefficient char-acterizing each R element, and<rn> its radius of order n, whereas Am
n represents the distribution of charges in the environment[8,9] In tetragonal symmetry, B2
2 terms are generally absent Here, the second order term B22;sO22;srepresents the fact that the two atomic positions of the R site have local orthorhombic symmetry, with the in-plane principal axes rotated by 90between the two sets, so that the total anisotropy has the tetragonal symmetry of the crystal structure Finally, the in-plane anisotropy is only determined by the higher order terms B4;c4 O4;c4 and B4;c6 O4;c6 [4] Note that higher order terms decrease very rapidly with increasing temperature[10,11], so that at room temperature and above, second order terms always dominate The assimilation of tetragonal symmetry to uniaxial symmetry is equivalent to neglecting higher-order terms and it becomes more valid as temperature is increased
A number of studies on single crystalline samples permitted the determination of CEF parameters in R2Fe14B with various R ele-ments[12e14] In particular, it was noted in these studies that the values of the parameters Am
n found in Nd2Fe14B give satisfactory account for the behaviour of compounds with other R elements (see Ref.[12])
3 A classical description of the properties of R2Fe14B compounds
Using a classical molecular field approach, the temperature dependence of the Fe magnetization and that of the R magnetiza-tion were derived in Ref [5] for the R2Fe14B compounds with R¼ Nd, Pr and Dy In addition, the exchange and CEF parameters were used to evaluate classical anisotropy coefficients, km
n, where the index n and m are the same as above[8] From the km
n values, the Ki anisotropy constants were obtained, where the order of the anisotropy constants is equal to 2i In the derivation, only the terms representative of uniaxial anisotropy were kept In-plane anisot-ropy terms were neglected for the reason explained above At any given temperature, all parameters characterizing the magnetic properties in a classical approach are known, and thefield depen-dence of the magnetization along a field applied in the plane perpendicular to the uniaxial axis,c, may be derived by minimi-zation of the total energy density expressed as:
ETẳ KFesin2wFeỵ K1Rsin2wRỵ K2Rsin4wRỵ K3Rsin6wR nRFe< MR>T< MFe>TcosðwFe wRÞ
Bapp< MR>TsinðwRÞ Bapp< MFe>TsinðwFeÞ (2)
where KFeis the second order anisotropy constant of Fe, K1R, K2Rand K3Rthe second, fourth and sixth-order anisotropy constants of the R atom (all expressed in J/m3), <MFe> and <MR>T are the finite
temperature values of the Fe and R magnetization (in A/m), nRFeis the associated molecularfield coefficient (a number multiplied by
m0in SI),wFeandwRare the angle of the Fe and R moments with respect toc, and Bapp is the applied magneticfield expressed in Tesla Such magnetization curves were obtained in Ref.[5] Calculating the magnetic properties of pseudo-ternary (ReR′)2(FeeCo)14B compounds
The RFeB-based magnets used in hybrid electric vehicles and wind turbines now contain heavy R elements, such as Dy or Tb, which partially substitute Nd, so as to increase magnetocrystalline anisotropy, and thus coercivity, at the elevated operating temper-atures (Top) which may reach 180C In addition, a fraction of Co is often substituted for Fe to increase the Curie temperature and in turn the R magnetocrytalline anisotropy at Top (the magneto-crystalline anisotropy at a given temperature is a function of the relative magnetization at that temperature, itself depending essentially on T/TC) These considerations imply that not only the magnetic properties of simple ternary compounds but also those of pseudo-ternary compounds, incorporating Fe and Co atoms on the one hand, and different R atoms on the other, should be calculated To calculate the magnetic properties of pseudo-ternary com-pounds, having general compositionðR1xR0xÞ2ðFe1yCoyÞ14B, the effect of Co on the magnetic properties must be evaluatedfirst The Curie temperature of a compound R2(Fe1yCoy)14B with non-magnetic R, may be expressed as:
TMẳ
1 yịTFeỵ yTCo ỵ
yịTFe yTCoị2ỵ 41 yÞyT2FeCo
q
(3)
where the index M in TMstands for transition metal, TFe, TCoand TFeCo are the Curie temperatures associated with FeeFe, CoeCo and FeeCo exchange interactions, respectively Gavigan et al showed that in R2(FeeCo)14B compounds, FeeCo interactions (TFeCo¼ 1025 K) are much stronger than FeeFe interactions (TFe¼ 565 K), and are as strong as CoeCo interactions (TCo¼ 1025 K)[15]
The Curie temperature in a compound where two elements, R and R0, are mixed, is easily derived from the expression obtained in the case where only one R element is present [7] It reads (neglecting ReR interactions as already indicated):
TC¼
TMỵ
T2Mỵ 41 xịT2
RMỵ 4xT2R0M
q
(4)
where TMis given by expression(3), x is the fraction of R0atoms substituted for R ones, TRR0ịMẳ nRR0ịM
CRR0ịCM q
, with CM, CRand CR0being the Curie constants associated with the M, R and R0atoms,
respectively For calculation of the Curie constants, it was assumed that there are 59.4$1027M atoms per m3and 8.5$1027R atoms per m3in the R2M14B compounds, the M effective moment was taken as (1y) 4mBỵ y 3.3mB, where the Fe and Co effective moments (4mB and 3.3mB) were taken from Ref.[6], and the trivalent R ion effective moments were used The nRM molecular field coefficients were taken from Ref.[7](M¼ Fe) and Ref.[6](M¼ Co)
(3)represented by a simple expression However, considering the similarities in the transition metal magnetic properties for all compounds in the R2M14B series, it is justified to identify the 3d anisotropy in all R2(FeeCo)14B compounds with the one found in the Y-based compound Hong et al.[16]determined the anisotropy and its temperature dependence in the Y2(FeeCo)14B compounds Note the anomalous behaviour observed: at low temperature, Co substitution initially leads to an increase in the 3d uniaxial anisotropy, whereas for y> 0.25, the anisotropy starts to decrease; Y2Co14B is a basal plane system This non-monotonous dependence of the 3d anisotropy upon Co substitution is indicative of prefer-ential occupancy by Co atoms of specific atomic sites in the tetragonal structure The increase in anisotropy occurring at low temperature is not preserved however above room temperature due to the decrease of KCowith increasing temperature, in contrast to the anomalous temperature dependence of KFein the R2Fe14B compounds, which increases with T, up to 300 K[18]
The temperature dependence of the R magnetization and that of the R anisotropy constants were calculated using the molecular field approach, with values of anisotropy constants derived from values of the CEF parameters given in Ref.[5] As a typical example, all derived parameters used for the calculation of the magnetiza-tion curves described below, are gathered inTable 1(for Fe and Co) andTable 2(for R atoms) for x¼ 0.25 and y ¼ 0.25
The expression used to evaluate thefield dependence of the magnetization was directly obtained from expression(2) It is:
Table
Magnetic parameters involved in the calculation of the 3d magnetic properties (Fe, Co) in R2M14B compounds, for x¼ 0.25 and y ¼ 0.25, at 300 K and 453 K <mFe(Co)>Tis
the value of the Fe (Co) magnetic moment at the considered temperature The other parameters are defined in the text
T (K) <mFe>T
(mB/atom)
<MFe>T
(106A/m) <(mmCo>T B/atom)
<MCo>T
(106A/m)
KM
(106J/m3)
300 2.07 1.14 1.34 0.74 1.08
453 1.91 1.05 1.24 0.68 1.08
Table
Magnetic parameters involved in the calculation of the rare-earth (R) magnetic properties in R2M14B compounds, for x¼ 025 and y ¼ 0.25, at 300 K and 453 K
<mR(R0)>Tis the value of the R(R0) magnetic moment at the considered temperature
The other parameters are defined in the text
T¼ 300 K T¼ 453 K
Nd Tb Dy Nd Tb Dy
<mR(R0)>T(mB/atom) 2.1 6.3 6.0 1.5 4.7 4.2
<MR(R0)>T(106A/m) 0.16 0.50 0.47 0.11 0.37 0.33
K1R(R0)(106J/m3) 3.7 11.6 6.7 1.9 5.8 3.2
K2R(R0)(104J/m3) 50 45 22 7.7 10 4
K3R(R0)(104J/m3) 10 3 2 1 0 0
(4)ETẳ KMsin2wMỵ K1Rsin2wRỵ K1R0sin2wR0ỵ K2Rsin4wR
ỵ K2R0sin4wR0ỵ K3Rsin6wRỵ K3R0sin6wR0
nRM< MR>T< MM>TcosðwM wRÞ nR0M< MR0>T< MM>TcosðwM wR0Þ
Bapp< MR>TsinðwRÞ Bapp< MR0>TsinðwR0Þ
Bapp< MM>TsinðwMÞ (5)
all terms have the same meaning as in expression(2), with the index M for the transition metal, the index R for thefirst rare-earth atom, Nd in the present case, and the index R0for the second rare-earth atom (Dy or Tb) The R and R0magnetization and anisotropy constants, in this expression(5), are affected by a coefficient equal to (1 x) for R atoms, and to x for R0atoms.
Calculation of the magnetic properties of pseudo-ternary com-pounds was performed at two temperatures, 300 K and 453 K respectively, the latter corresponding to the typical maximum operating temperature encountered in hybrid electric vehicles and wind turbines No further adjustment of the calculated curves to approach experimental curves was applied
The calculated magnetization curves in (Nd1xTbx)2Fe14B and (Nd1xDyx)2Fe14B at 300 K are presented inFig Thefield de-pendences of the magnetization in the ternary compounds are in
fair agreement with literature data[3e5,12] Qualitatively, the in-crease in anisotropy induced by the introduction of Tb or Dy manifests itself as a reduction in the slope characterizing the magnetization variation underfield However, as noticed in Ref.[5], in such ferrimagnetic materials where strong non-collinearity be-tween the magnetic moments is induced by the applied magnetic field, the slope of the magnetization variation is not directly related to the anisotropy constant
The calculations were extended to large magneticfields above 100 T (Fig 1, right) In both series of compounds, full saturation is reached in magneticfield of the order of 150 T or above At satu-ration, the Tb or the Dy moments, which couple antiparallel to the Fe moments in zero appliedfield, under the effect of the exchange field, have rotated and become aligned with the field The field at which saturation is reached is thus representative of TbeFe or DyeFe interactions, amounting to values of the order of 200 T and 150 T, respectively The High Field Free Powder method (HFFP), developed by the Amsterdam group in the 1990s, constitutes an experimental approach to obtain the strength of exchange coupling
[19] With the development of magneto-optic measurements in high pulsed magneticfields[20,21], the possible use of the HFFP method to the present compounds could be explored
The calculated magnetization curves in (Nd1xTbx)2(Fe 1-yCoy)14B and (Nd1xDyx)2(Fe1yCoy)14B at 300 K are presented in
Fig The continuous black lines in thesefigures represent the Fe
Fig Calculated magnetization curves of (NdeTb)2(FeeCo)14B (top) and (NdeDy)2(FeeCo)14B (bottom) in an applied magneticfield of up to 25 T The black lines correspond to Co
(5)compound and the continuous blue lines represent compounds containing cobalt The blue lines are always below the black lines due to the reduced magnetization resulting from Co substitution The dashed blue lines represent the calculated magnetization of a hypothetical compound having the same Co content as the com-pound represented by the continuous blue lines, but in which the 3d magnetic interactions would be the same as in a compound containing only Fe For any given composition, the continuous blue line is always above the dashed blue line This illustrates the fact that the thermally induced decrease of magnetization is reduced in Co compounds due to the higher values of the Curie temperature Note also that magnetic saturation in Co containing compounds requires a stronger magnetic field, i.e the magnetocrystalline anisotropy is increased The enhanced magnetic interactions introduced by the presence of Co, at a given T, result in a relatively higher value of the rare-earth magnetic moment and, subsequently of the R anisotropy, which is a function of the R moment, to some power
M(H) curves at 453 K are compared to room temperature curves in Fig At 453 K, the saturated magnetization of compounds containing Co is above the magnetization of Co-free compounds The reduced temperature dependence of the magnetization more than compensates the fact that the zero Kelvin magnetization is reduced by Co substitution This illustrates the interest of Co sub-stitution for high temperature applications
5 Conclusions
Meanfield calculations of the magnetic properties of pseudo-ternary R2M14B compounds illustrate how the magnetic anisot-ropy of such systems may be adjusted by playing with rare-earth and Co substitution These calculations involve a limited number of parameters, applied to all compounds The results presented here are directly applicable to the analysis of magnets in which R and Co substitution is made in the starting alloy, and in principle they may feed into micro-magnetic modelling of recently devel-oped diffusion processed magnets in which R substitution occurs at the surface of individual Nd2Fe14B grains
Acknowledgements
This paper is based on results obtained from the“Development of magnetic material technology for high-efficiency motors” pro-gram commissioned by the New Energy and Industrial Technology Development Organization (NEDO) of Japan
The paper is dedicated to the memory of Peter Brommer, a highly respected scientist with whom some of us (DG and NMD) have benefited from scientifically fruitful and friendly exchanges, in particular during common visits to Vietnam
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